Finite Group Databasebeta

← #29605

Subgroups — #29605

Order 256.

634 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#536C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#451C211yesmaximal, characteristic, complemented
4#2616C211yesmaximal, characteristic
5#2036C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#2001C212yesmaximal, complemented
9#1825C211yesmaximal, characteristic, complemented
10#1787C211yesmaximal, characteristic, complemented
11#1780C211yesmaximal, characteristic
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C22×Q16C2212yes—
15#536—24no—
16#536—24no—
17C2×C4⋊Q8—21no—
18C2×#124—21no—
19C2×#123—22no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2212yes—
22C2×C22⋊Q8C2212yes—
23C2×C22⋊Q8C2212yes—
24C2×#117C2211yescharacteristic
25#485—24no—
26#451—24no—
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2212yes—
37C2×C22⋊C8C2212yes—
38#395—21no—
39#394—22no—
40#394—21no—
41#380C2211yescharacteristic
42#380C2212yes—
43#340C2212yes—
44#340C2212yes—
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—42no—
48C2×#41—42no—
49C22×Q8—21no—
50C22×Q8—22no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8D414yes—
54C22×Q8D411yescharacteristic
55C22×Q8C2311yescharacteristic
56C22×Q8C2312yes—
57C22×D4D412yes—
58C22×D4D411yescharacteristic
59C23×C4—21no—
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62C2×Q16—24no—
63C2×Q16—42no—
64C2×Q16—42no—
65C22×C8C2312yes—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C4⋊Q8—28no—
69#124—28no—
70#123—216no—
71C22⋊Q8—24no—
72C22⋊Q8—24no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8—42no—
78C22⋊Q8—42no—
79#117—22no—
80#117D414yes—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C42—21no—
97#107—22no—
98#106—22no—
99#103—24no—
100#103—22no—
101#103—22no—
102#103—42no—
103#103—42no—
104C22⋊C8—22no—
105C22⋊C8—22no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108#95—22no—
109#95—21no—
110C24C2×D412yes—
111C24C2×D411yescharacteristic
112#41—48no—
113#41—44no—
114#41—44no—
115C2×Q8—24no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—42no—
127C2×Q8—48no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8C2×D414yes—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—42no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—44no—
153C22×C4C2×D412yes—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D412yes—
156C22×C4C2×D412yes—
157C22×C4C2411yescharacteristic
158Q16—48no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C4⋊C4—28no—
167C4⋊C4—28no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C4⋊C4C2×D412yes—
177C4⋊C4C2×D412yes—
178C22⋊C4—28no—
179C22⋊C4—28no—
180C22⋊C4—24no—
181C22⋊C4—44no—
182C22⋊C4—42no—
183C22⋊C4—81no—
184C22⋊C4—82no—
185C42—24no—
186C23—22no—
187C23—21no—
188C23—22no—
189C23—21no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23#12012yes—
198C23C22×D411yescharacteristic
199Q8—24no—
200Q8—24no—
201Q8—48no—
202Q8—44no—
203Q8—42no—
204Q8—42no—
205Q8—42no—
206Q8—41no—
207Q8—41no—
208D4—44no—
209D4—48no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—48no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4#12012yes—
239C2×C4C22×D411yescharacteristic
240C2×C4C22×D411yescharacteristic
241C2×C4H(5,𝔽2)12yes—
242C2×C4H(5,𝔽2)12yes—
243C8—42no—
244C8—41no—
245C8—41no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22C2×#12011yescharacteristic
259C22#53412yes—
260C22#57711yescharacteristic
261C22#57812yes—
262C22#57811yescharacteristic
263C4—22no—
264C4—42no—
265C4—44no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—82no—
270C4—81no—
271C4#53412yes—
272C4#53511yescharacteristic
273C4#53511yescharacteristic
274C2—42no—
275C2—82no—
276C2#261311yescharacteristic
277C2#301111yescharacteristic
278C2#301311yescharacteristic
279C2#320112yes—
280C2#320212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#485C2semidirect
3. C2×#451C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #2001C2semidirect
8. #2001C2semidirect
9. #1825C2semidirect
10. #1787C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#5362
C2×#4852
C2×#4512
#26162
#20362
#20352
#20012
#18252
#17872
#17802